Universal

The four sorts (G, T, R, C) classify all well-formed expressions without overlap or gap. Every well-formed expression belongs to exactly one sort.

The abstract and embed operations form a dual pair: abstracting an embedded expression recovers the original, and embedding an abstracted expression recovers the original, up to canonical equivalence.

The Erlangen Program hierarchy applied to UL captures all geometrically meaningful distinctions between expressions. No finer classification is needed.

Any concept expressible in any natural or formal language can be represented as a well-formed UL expression using the five primitives and eleven operations.

Universal Language can describe its own syntax, semantics, and proof system entirely within its own formalism. UL is self-describing.

Every invertible well-formed expression has a unique inverse under the invert operation. The set of invertible expressions forms a group under composition.